Download Electromagnetic waves

Physics 272
November 25
Fall 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_fall_272_uhm.html
Prof. Philip von Doetinchem
[email protected]
Phys272 - Fall 14 - von Doetinchem - 105
Electromagnetic waves
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What is light?
→ electromagnetic wave
→ electromagnetism is needed
(not the complete story → QFT)
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Time varying magnetic field creates electric field
Time varying electric field creates magnetic field
→ sustain each other and create an
electromagnetic wave that propagates through
space
Phys272 - Fall 14 - von Doetinchem - 106
Electromagnetic waves
Source: http://en.wikipedia.org/wiki/Light
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Electromagnetic waves carry energy and momentum
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Electric and magnetic fields in sinusoidal waves have defined frequency
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Infrared, visible light, UV, X-ray, -ray, etc. all follow the same principle,
but at different wavelength
Electromagnetic waves do not require medium (like mechanical waves)
Phys272 - Fall 14 - von Doetinchem - 107
Electricity, magnetism, and light
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Maxwell's equation relate magnetic and electric field
Moving charges produce both electric and magnetic
fields
An electromagnetic wave is formed when a charge
accelerates
We never spoke about how fast a magnetic field can
be measured at a certain distance after a charge
starts moving:
–
Electromagnetic waves do not travel with infinite speed
Phys272 - Fall 14 - von Doetinchem - 108
Generating electromagnetic radiation
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Look at oscillating L-C circuit
Transformation of the LC circuit into a conducting
rod:
→ this is still an oscillating LC circuit
Phys272 - Fall 14 - von Doetinchem - 109
Generating electromagnetic radiation
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What do the electric and magnetic fields look like:
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In initial configuration fields are contained in capacitor and inductor
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In the rod configuration the electric and magnetic field overlap
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Current is moving up and down the rod
→ charge in “capacitor” and current in “inductor” are changing with
time
→ electric and magnetic field change with time
→ electric and magnetic field propagate with finite velocity
→ electromagnetic wave
Phys272 - Fall 14 - von Doetinchem - 110
Hertzian dipole
Source: http://de.wikipedia.org/wiki/Hertzscher_Dipol
Change of electric field over time
Phys272 - Fall 14 - von Doetinchem - 111
Plane electromagnetic waves
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Electromagnetic waves are transverse waves with an electric
and a magnetic component
→ similar to waves on a string
Make the following assumption:
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Electric and magnetic field configuration with wave-like behavior
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Electric field has only a y component
–
Magnetic field has
only a z component
–
Both move in x
direction with
velocity c
We have to test if
this assumption is
consistent with
Maxwell's equation
Phys272 - Fall 14 - von Doetinchem - 112
Plane electromagnetic waves
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Why perpendicular?
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Faraday's law:
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Integration path
perpendicular to electric
field: no contribution
Propagation velocity
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Magnetic flux change:
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Magnetic field component in z direction is crucial to comply to Faraday's law
Phys272 - Fall 14 - von Doetinchem - 113
Plane electromagnetic waves
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In a very similar way:
use Ampere's law (no conduction current):
any components of E and B not perpendicular to each
other and not perpendicular to the direction do not play a
role to fulfill Maxwell's equation
→ Electric and magnetic fields must be perpendicular
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To fulfill Maxwell's equations → wave must also fulfill:
Propagation velocity
Phys272 - Fall 14 - von Doetinchem - 114
Key properties of electromagnetic waves
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Electromagnetic waves fulfill the general wave equation for both electric and
magnetic fields (not discussed here):
Wave is transverse:
–
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Electric field and magnetic field are perpendicular to the direction of the wave
Magnitudes of E and B are related by the propagation velocity: E=cB
In vacuum (or if medium is not changing) the electromagnetic wave is traveling
at a constant velocity
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Electromagnetic waves do not need a medium
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Simple plane waves can be generalized
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E and B fields can be superposed
–
For each superposition the simple principles of the plane electromagnetic waves apply
–
Also wave packets or sinusoidal waves fulfill Maxwell's equations
Phys272 - Fall 14 - von Doetinchem - 115
Sinusoidal electromagnetic waves
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At any instant the fields are uniform over any plane
perpendicular to the direction of propagation with speed c
The entire pattern
travels in the direction
of propagation
Electric and magnetic field
are still transverse to the
propagation direction at
any instant
In a small region of space at great distance from source
→ electromagnetic waves can be treated as plane waves
Phys272 - Fall 14 - von Doetinchem - 116
Fields of a sinusoidal waves
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Electric and magnetic fields oscillate in phase
The fields are at the
same time at
maximum, zero,
and minimum
Vector product of
electric field and
magnetic field always
points in the propagation direction
Electromagnetic wave can be described by wave function
depending on location and time
k is the wave number: 2/
Phys272 - Fall 14 - von Doetinchem - 117
Fields of a sinusoidal waves
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The electric field and magnetic field amplitudes are
related by:
The sign in front of t denotes the direction of the
wave:
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Negative sign: positive x direction
–
Positive sign: negative y direction
Phys272 - Fall 14 - von Doetinchem - 118
Why are magnetic and electric field in phase?
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Assume arbitrary phase angle:
Relation between electric and magnetic field using
Faraday's law:
Phys272 - Fall 14 - von Doetinchem - 119
Why are magnetic and electric field in phase?
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Relation between electric and magnetic field using
variation of B is
Faraday's law:
small for small x
calculate magnetic
flux through area
Phys272 - Fall 14 - von Doetinchem - 120
Why are magnetic and electric field in phase?
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Use our approach for the wave function assuming a
phase angle:
To make these equations equal at all times we have
to require
Phys272 - Fall 14 - von Doetinchem - 121
Electromagnetic waves in matter
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The wave speed changes in matter
(we assume non-conducting dielectric)
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We have to use the dielectric constant and the permittivity of the material:
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Electromagnetic waves can never travel faster than the speed of light in vacuum
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Definition of index of refraction n of a material
(→ optics)
Careful:
–
dielectric “constant” depends on frequency
–
for high frequencies materials cannot be polarized as fast as for constant electric
fields
→ reduces dielectric “constant”
Phys272 - Fall 14 - von Doetinchem - 122
Energy and momentum in electromagnetic waves
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Electromagnetic waves transport energy:
–
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Microwave ovens, radio transmitters, lasers for eye
surgery, …
Combined energy density of electric and magnetic
components:
In vacuum electric field and magnetic field carry
half of the total energy density
Phys272 - Fall 14 - von Doetinchem - 127
Electromagnetic energy flow and the Poynting vector
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Energy is required to establish electric and magnetic fields
→ electromagnetic waves transport energy from one region to the other
Definition:
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energy transferred per unit time per unit cross-sectional area
–
or power per unit area
In a certain time the wave moves in space with the propagation velocity
→ energy in a particular region:
Phys272 - Fall 14 - von Doetinchem - 128
Electromagnetic energy flow and the Poynting vector
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Poynting vector is a
vector quantity:
it points in the direction
of the propagation of the wave
and depends on time
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SI unit: 1 W/m2
John Henry Poynting
(1852-1914)
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Total energy flow per unit time:
Phys272 - Fall 14 - von Doetinchem - 129
Electromagnetic energy flow and the Poynting vector
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Frequency of electromagnetic
waves is fast → average Poynting
vector value (intensity)
For our sinusoidal wave:
zero on average
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The intensity for a sinusoidal wave is exactly half of the
maximum value:
This is what we sense when looking at, e.g., light from the
sun
→ variations are too fast to be noticeable for us
Phys272 - Fall 14 - von Doetinchem - 130
Electromagnetic momentum flow and radiation pressure
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Electromagnetic waves also carry momentum
(→ deeper explanation requires quantum physics)
Electromagnetic wave's momentum is transferred to
a surface
Transferred momentum
per time equals the
force on the surface
Sun and stars
create radiation
pressure that effects the
surrounding material
→ star forming regions
Massive Star Forming Region DR21 in Infrared
Credit: A. Marston (ESTEC/ESA) et al., JPL, Caltech, NASA
Phys272 - Fall 14 - von Doetinchem - 131
Standing electromagnetic waves
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Electromagnetic waves
can be reflected on
surfaces
–
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Dielectrics or conductors
can serve as reflectors
Superposition principle of
electric and magnetic
fields also applies to
electromagnetic waves
Superposition of incident
and reflected wave forms
a standing wave
Electric force is
conservative → it is not
possible to do work on a
test charge like that:
Phys272 - Fall 14 - von Doetinchem - 134
Additional Material
Phys272 - Fall 14 - von Doetinchem - 135
Energy in a sinusoidal wave
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A radio station on the
earth's surface emits
a sinusoidal wave with
average total power of
50kW
Assume that transmitter radiates equally in all
directions
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Area of hemisphere:
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Intensity:
Phys272 - Fall 14 - von Doetinchem - 136
Energy in a sinusoidal wave
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A radio station on the earth's
surface emits a sinusoidal
wave with average total power
of 50kW
Assume that transmitter radiates
equally in all directions
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Electric field amplitude:
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Magnetic field amplitude:
magnetic field associated with electromagnetic wave is extremely small compared to
what we saw before
→ it is easier to have device sensitive to the electric field
Phys272 - Fall 14 - von Doetinchem - 137
Solar sail
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Solar sails try to make use out of the radiation
pressure as a propulsion system
Usage of sails big as football fields should catch the
radiation pressure by the sun
Concept is proven, but
not yet part of real
spacecrafts
Source: http://en.wikipedia.org/wiki/Solar_sail
Phys272 - Fall 14 - von Doetinchem - 138